Adaptive Mesh Generation Research Articles

An automatic strategy for adaptive tetrahedral mesh generation

This paper introduces a new automatic strategy for adaptive tetrahedral mesh generation. A local refinement/derefinement algorithm for nested triangulations and a simultaneous untangling and smoothing procedure are the main involved techniques. The mesh generator is applied to 3-D complex domains whose boundaries are projectable on external faces of a meccano approximation composed of cuboids. The domain surfaces must be given by a mapping between meccano surfaces and object boundary. This mapping can be defined by analytical or discrete functions. At present, we have fixed mappings with orthogonal, cylindrical and radial projections, but any other one-to-one projection may be considered. The mesh generator starts from a coarse and valid hexahedral mesh that is obtained by an admissible subdivision of the meccano cuboids. The automatic subdivision of each hexahedron into six tetrahedra produces an initial tetrahedral mesh of the meccano approximation. The main idea is to construct a sequence of nested meshes by refining only those tetrahedra with a face on the meccano boundary. The virtual projection of meccano external faces defines a valid triangulation on the domain boundary. Then a 3-D local refinement/derefinement is carried out so that the approximation of domain surfaces verifies a given precision. Once this objective is reached, those nodes placed on the meccano boundary are really projected on their corresponding true boundary, and inner nodes are relocated using a suitable mapping. As the mesh topology is kept during node movement, poor quality or even inverted elements could appear in the resulting mesh; therefore, we finally apply a mesh optimization procedure. The efficiency of the proposed technique is shown with several applications to complex objects.

New high quality adaptive mesh generator utilized in modelling plasma streamer propagation at atmospheric pressures

A new adaptive mesh generator has been developed and used in the analysis of high-pressure gas discharges, such as avalanches and streamers, reducing computational times and computer memory needs significantly. The new adaptive mesh generator developed, uses normalized error indicators, varying from 0 to 1, to guarantee optimal mesh resolution for all carriers involved in the analysis. Furthermore, it uses h- and r-refinement techniques such as mesh jiggling, edge swapping and node addition/removal to develop an element quality improvement algorithm that improves the mesh quality significantly and a fast and accurate algorithm for interpolation between meshes. Finally, the mesh generator is applied in the characterization of the transition from a single electron to the avalanche and streamer discharges in high-voltage, high-pressure gas discharges for dc 1 mm gaps, RF 1 cm point–plane gaps and parallel-plate 40 MHz configurations, in ambient atmospheric air.

An assessment by a commercial software of the accuracy of electromagnetic finite element simulators in the presence of metamaterials

PurposeThe paper's purpose is to deduce additional information on the accuracy of finite element simulators for electromagnetic problems involving effective models of metamaterials.Design/methodology/approachThe objective is achieved by solving, with a well known commercial simulator, many different configurations of two types of electromagnetic problems: a free‐space scattering by a sphere and a waveguide discontinuity problem. Such problems are known to be able to point out the difficulties of numerical simulators. On the other hand, they are representative of two important classes of problems and can provide indications on what can happen in other cases.FindingsThis analysis confirms that the numerical errors can be important just in close proximity of the interface between metamaterials and standard media. Small values of loss tangents can be sufficient to obtain very accurate results. Adaptive mesh generators should not be used in the presence of negligible values of the loss tangents. For more uniform meshes the results are satisfactory, with sufficiently fine meshes. When the magnitude of the real parts of the effective dielectric permittivity of a metamaterial and of the adjacent standard media are significantly different, the accuracy is satisfactory in any case.Research limitations/implicationsThe results are obtained by considering problems of two types. There is no guarantee that all the deductions apply to other models.Practical implicationsTo design practical devices involving metamaterials reliable electromagnetic simulators are necessary. The reported results seem to indicate that it is possible to adopt some countermeasures against the possible lack of accuracy of finite element simulators in the presence of effective models of metamaterials.Originality/valueFor the first time, to the best of authors' knowledge, an extensive analysis on the accuracy of finite element simulators for critical problems involving metamaterials has been carried out. Some simple suggestions to improve their reliability in these cases are provided.

Adaptive and anisotropic mesh strategy for thin shell problems. Case of inhibited parabolic shells

The aim of this paper is to show the reliability of an adaptive and anisotropic mesh procedure for thin shell problems. We consider singular perturbation problems only for parabolic shells whose behavior is described by the Koiter model. The corresponding system of equations, which depends on the relative thickness ε of the shell, is elliptic except at the limit for ε = 0 where it is parabolic. In a first part of this paper, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. In a second part, numerical computations are performed using a finite element software coupled with an adaptive anisotropic mesh generator. This technique enables to approach accurately the singularities and the layers predicted by the theory especially for very small values of the thickness. The efficiency of such a procedure in comparison with uniform meshes is put in a prominent position.

Singular perturbations generating complexification phenomena for elliptic shells

This paper deals with elliptic shell problems using the Koiter shell model. When the shell is well-inhibited, the limit membrane problem satisfies the Shapiro–Lopatinskii condition and we have a classical singular perturbation problem. In a previous paper, the existence of two kinds of singularities was put in a prominent position for this kind of problem. Conversely, for ill-inhibited shells (when a part of the boundary is free), the limit problem does not satisfy the Shapiro–Lopatinskii condition. Complexification phenomenon appears when the thickness approaches zero, leading to large oscillations corresponding to a new kind of instability on the free boundary. To complete the theoretical analysis, numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. This enables us to visualize precisely the singularities and the instabilities predicted by the theory with only a small number of elements.

Numerical Wave Tanks Based on Finite Element and Boundary Element Modeling

This work forms part of an investigation into the nonlinear interaction between steep (but not overturning) transient waves and flared structures, using a coupled finite element and boundary element model. The use of a coupled approach is based on consideration of the relative strengths and weaknesses of the finite element (FE) and boundary element (BE) methods when implemented separately (e.g., efficiency of computation versus complexity of adaptive mesh generation). A FE model can be used to advantage away from the body, where the domain is regular, and a BE discretization near the body where the moving mesh is complex. This paper describes the aspects of the FE and BE models which have been developed for this analysis, each based on the use of quadratic isoparametric elements implemented in a mixed Eulerian–Lagrangian formulation. Initially, the two approaches have been developed side by side, in order to ensure the use of robust components in the coupled formulation. Results from these methods are obtained for a series of test cases, including the interaction of an impulse wave with a circular cylinder in a circular tank, and nonlinear diffraction by a cylinder in a long tank.

Variational method for adaptive mesh generation

A variational method is suggested for generating adaptive grids composed of hexahedral cells. The method is based on the minimization of a functional written on a manifold in a space whose variables are usual spatial coordinates in a physical domain and the components of a monitor vector function. A grid is constructed in the manifold, and its projection onto the physical domain yields an adaptive grid. Examples of adaptive grid generation are given.

Automatic adaptive FE analysis of thin‐walled structures using 3D solid elements

AbstractIn this study, a new automatic adaptive refinement procedure for thin‐walled structures using 3D solid elements is suggested. This procedure employs a specially designed superconvergent patch recovery (SPR) procedure for stress recovery, the Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation, a new refinement strategy for new element size prediction and a special mesh generator for adaptive mesh generation. The proposed procedure is different from other schemes in such a way that the problem domain is separated into two distinct parts: the shell part and the junction part. For stress recovery and error estimation in the shell part, special nodal coordinate systems are used and the stress field is separated into two components. For the refinement strategy, different procedures are employed for the estimation of new element sizes in the shell and the junction parts. Numerical examples are given to validate the effectiveness of the suggested procedure. It is found that by using the suggested refinement procedure, when comparing with uniform refinement, higher convergence rates were achieved and more accurate final solutions were obtained by using fewer degrees of freedoms and less amount of computational time. Copyright © 2008 John Wiley & Sons, Ltd.

Adaptive tetrahedral mesh generation by constrained Delaunay refinement

AbstractThis paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three‐dimensional space. The main issues are the mesh quality and mesh size, which directly affect the accuracy of the numerical solution and the computational cost. Two basic problems need to be resolved, namely boundary conformity and field points distribution. The proposed method utilizes a special three‐dimensional triangulation, so‐called constrained Delaunay tetrahedralization to conform the domain boundary and create field points simultaneously. Good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains. In addition, an isotropic size field associated with the numerical solution can be supplied; the field points will then be distributed according to it. Good mesh size conformity can be achieved for smooth sizing informations. The proposed method has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance. Copyright © 2008 John Wiley & Sons, Ltd.

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvement of meshes in a broad range of applications.

Finite element model for linear-elastic mixed mode loading using adaptive mesh strategy

An adaptive mesh finite element model has been developed to predict the crack propagation direction as well as to calculate the stress intensity factors (SIFs), under linear-elastic assumption for mixed mode loading application. The finite element mesh is generated using the advancing front method. In order to suit the requirements of the fracture analysis, the generation of the background mesh and the construction of singular elements have been added to the developed program. The adaptive remeshing process is carried out based on the posteriori stress error norm scheme to obtain an optimal mesh. Previous works of the authors have proposed techniques for adaptive mesh generation of 2D cracked models. Facilitated by the singular elements, the displacement extrapolation technique is employed to calculate the SIF. The fracture is modeled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. The SIFs values for two different case studies were estimated and validated by direct comparisons with other researchers work.

A finite element model of electromigration induced void nucleation, growth and evolution in interconnects

We describe a two-dimensional finite element method designed to model the nucleation, growth and evolution of voids in polycrystalline interconnects. Our computations extend earlier models by accounting in detail for the effects of grain boundaries and interfaces in the solid. In particular, the model includes the effects grain boundary sliding, grain boundary diffusion, grain boundary migration and surface diffusion, as well as elastic deformation and electric current flow within the grains themselves. Front tracking and adaptive mesh generation are used to follow changes in the grain structure and void shape. The method is used to solve representative boundary value problems to illustrate its capabilities.

Accurate numerical models for simulation of radiation events in nano-scale semiconductor devices

Space and ground-level electronic equipment with devices of nanometer scale and extremely high densities may be strongly affected by ionizing radiation. Accurate numerical models are necessary for better analysis of radiation effects and design of radiation tolerant devices. Numerical models and adaptive dynamic 3D mesh generation presented in this paper enable efficient simulations of transient semiconductor device response to realistic multi-branched track, produced by ionizing particle and nuclear reactions. A description of the numerical models, adaptive 3D meshing approach, and efficient solution method for large 3D problems using unstructured meshes is provided. The developed approach efficiently uses computer resources, which makes possible solution of transient 3D multi-branched ion strike problems with 100,000 mesh nodes within 512MB computer memory.

Computing singular perturbations for linear elliptic shells

This paper is devoted to the theoretical and numerical study of singular perturbation problems for elliptic inhibited shells. We present a reduction of the classical membrane equations to a partial differential equation with respect to the bending displacement, which is well adapted to the study of singularities of the limit problem. For a discontinuous loading or when the boundary of the loading domain presents corners, we put in a prominent position the existence of two kinds of singularities. One of them is not classical; it reduces to a logarithmic point singularity at the corner of the loading domain. To finish numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. They enable to visualize precisely the singularities predicted by the theory with only a very small number of elements.

Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging: I

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.

Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging: II

In part I (Guven et al 2007 Inverse Problems 23 1115–33), we analysed the error in the reconstructed optical absorption images resulting from the discretization of the forward and inverse problems. Our analysis led to two new error estimates, which present the relationship between the optical absorption imaging accuracy and the discretization error in the solutions of the forward and inverse problems. In this work, based on the analysis presented in part I, we develop new adaptive discretization schemes for the forward and inverse problems in order to reduce the error in the reconstructed images resulting from discretization. The proposed discretization schemes lead to adaptively refined composite meshes that yield the desired level of imaging accuracy while reducing the size of the discretized forward and inverse problems. We present numerical experiments to validate the error estimates developed in part I and show the improvement in the accuracy of the reconstructed optical images with the new adaptive mesh generation algorithms.

Adaptive hexahedral mesh generation based on local domain curvature and thickness using a modified grid-based method

This paper presents an adaptive hexahedral mesh generation algorithm for three-dimensional models. It is based on the conventional grid-based method and geometry-adaptive technique to generate uniformly sized or locally refined hexahedral meshes. The first step is to construct the refinement information fields of the source points and the corresponding elements according to the surface curvature of the solid model. A thickness refinement criterion is then used to construct the thickness-based refinement information field of elements from digital topology. Secondly, the element refinement fields are integrated and modified according to the conformal refinement templates proposed in this paper, and used as a metric to generate an initial grid structure enveloping the solid model completely. In order to ensure the conformity of the generated mesh, a set of new refinement templates are proposed, and the corresponding modes for converting arbitrary refinement fields to the templates is established. Especially two kinds of new transition templates, which avoid the propagation of the refinement field, are added. Hexahedral element meshes capturing all the geometry characters of the solid models are generated. The effectiveness and robustness of the algorithm are tested through the applications in several complex three-dimensional models.

Two‐dimensional adaptive mesh generation

AbstractIn this paper, a two‐dimensional adaptive elliptic mesh generation system, derived from the Ryskin and Leal (RL) orthogonal mesh generation system based on the orthogonal condition (orthogonality) and the cell area equal‐distribution principle (adaptivity), is presented. The proposed generation system takes into account not only the mesh orthogonality and adaptivity but also the mesh smoothness by adopting a method that the distortion functions is determined by both the scale factors and the averaged scale factors of the constant mesh lines. Examples and application show that the proposed generation system is effective and easy to use. Copyright © 2007 John Wiley & Sons, Ltd.

Boundary and interior layers in turbulent thermal convection in cylindrical containers

To simulate turbulent Rayleigh-Benard Convection (RBC) in cylindrical containers a high-order finite volume method based on the Chorin ansatz was developed. Using cylindrical staggered grids with about 107 nodes clustered in vicinity of the boundary, we performed simulations of turbulent RBC in wide cylindrical containers of the aspect ratios 5 and 10 for Rayleigh numbers from 105 to 109. The numerical method including the von Neumann stability and the adaptive mesh generation are discussed together with the simulation results. The boundary and interior layers structures and the turbulent background contribution to heat transport are investigated.

重み付き評価型の学習機能を有するアダプティブ・メッシング手法

This paper presents an adaptive mesh generation with parameterized learning. The present method does not need to perform iterative processes of field analysis in contrast with the conventional adaptive meshing methods. The present method evaluates mesh qualities for each element by means of evaluation function, which is weighted linear combination of shape and area of elements, distance from material corners and so on. The element which has the worst value of the evaluation function is divided into a few elements according to its shape, and this procedure is repeated until the prescribed number of elements is obtained. By using the simple Genetic Algorithm (sGA), the weighting coefficients are optimized through learning with example models such that the resultant mesh has the lowest numerical error.The good mesh can be obtained without time-consuming computation, since the weight values for the mesh features are learned by the sGA. The present method would allow us to realize effective design and development of electromagnetic machine and devices.